No abstract

Let G = K ⋉ R n , where K is a compact connected subgroup of O(n) acting on R n by rotations. Let g ⊃ k be the respective Lie algebras of G and K, and pr : g * −→ k * the natural projection. For admissible coadjoint orbits, which is called the Corwin-Greenleaf multiplicity function. Let π ∈ G and τ ∈ K be the unitary representations corresponding, respectively, to O G and O K by the orbit method. In this paper, we investigate the relationship between n(O G , O K ) and the multiplicity m(π, τ ) of τ in the restriction of π to K. If π is infinite-dimensional and the associated little group is connected, we show that n(O G , O K ) = 0 if and only if m(π, τ ) = 0. Furthermore, for K = SO(n), n ≥ 3, we give a sufficient condition on the representations π and τ in order that n(O G , O K ) = m(π, τ ).